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Weak and strong solutions of equations of compressible
magnetohydrodynamics
Xavier Blanc, Bernard Ducomet
To cite this version:
Xavier Blanc, Bernard Ducomet. Weak and strong solutions of equations of compressible
magnetohy-drodynamics. Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, 2016,
�10.1007/978-3-319-10151-4_72-1�. �hal-01201867�
Weak and strong solutions of equations of compressible
magnetohydrodynamics
Xavier Blanc
1, Bernard Ducomet
21 Univ. Paris Diderot, Sorbonne Paris Cit´e,
Laboratoire Jacques-Louis Lions, UMR 7598, UPMC, CNRS, F-75205 Paris, France
2 CEA, DAM, DIF,
D´epartement de Physique Th´eorique et Appliqu´ee, F-91297 Arpajon, France
1
Basic equations of magnetohydrodynamics
A plasma is a mixture of ions, electrons and neutral particles. At the macroscopic level, beyond the kinetic description necessary in rarefied situations, dynamics of dense plasmas is governed by the interaction between the fluid components and the electromagnetic fields. When dealing with collisional plasmas [64] [67], it is often appropriate to use one-fluid approximations which are much simpler than a complete kinetic theory and relevant for a number of applications (astrophysics [20] [89] [100] [104], plasma physics [25] [27], electrometallurgy [45] etc...).
Moreover a further approximation is obtained when one studies the interaction between electromag-netic fields and a electric-conducting fluid with electrical effects negligeable compared to magelectromag-netic effects. This simplified scheme is called magnetohydrodynamics (MHD). It unifies classical fluid dynamics and magnetism of continuum media in a coupled way: electromagnetic fields induce currents in the moving fluid which produce forces which in turn modify the electromagnetic fields. If one considers a compressible heat-conducting fluid, the equations governing the system will be the compressible Navier-Stokes-Fourier system coupled to the set of Maxwell’s equations through suitable momenta and energy sources.
1.1
Modelling of the magnetic field
Recall that the electromagnetic field is governed by the Maxwell system [70] decomposed into • the Amp`ere’s law
∂tD + j = curlxH, (1)
• the Coulomb’s law
divxD = ̺c, (2)
• the Faraday’s law
∂tB + curlxE = 0, (3)
• the Gauss’s law
where D is the electric induction, B is the magnetic induction, E is the electric field, H is the magnetic field, ̺c is the density of electric charges and j the electric current. We suppose that these quantities are
linked by the constitutive relations
B = µH and D = εE, (5)
where µ, the magnetic permeability and ε the dielectric permittivity are positive quantities, generally depending on the electromagnetic field. Furthermore one observes that the electric density charge ̺cand
the current density j are interrelated through the electric charge conservation
∂t̺c+ divxj = 0, (6)
however we assume in the following that electric neutrality holds: then ̺c= 0.
Finally we suppose that the magnetic induction vector B is related to the electric field E and the macroscopic fluid velocity u via Ohm’s law
j = σ(E + u × B), (7)
where the electrical conductivity σ = σ(̺, ϑ, H) of the fluid is a positive quantity.
Now the magnetohydrodynamic approximation [12] [66] [98] consists in neglecting the term ∂tD in (1)
and we obtain
j = curlxH. (8)
This approximation corresponds to a strongly non-relativistic regime and has been justified in the 2-dimensional case by S. Kawashima and Y. Shizuta in [63], and more recently in 3-2-dimensional space by S. Jiang and F. Li (see [57] for isentropic flows and [58] for the general case).
Accordingly, equation (3) can be written in the form
∂tB + curlx(B × u) + curlx 1 σ curlx 1 µ B = 0, (9) where µ = µ(|H|) > 0. Setting M(s) = Z s 0 τ ∂τ(τ µ(τ )) dτ, (10) equation (3) becomes ∂tM(|H|) + j · E = divx(H × E) . (11)
Supposing now that we study the previous system in a bounded region Ω ⊂ R3, we add boundary
conditions
B · n|∂Ω= 0, E × n|∂Ω= 0. (12)
and initial conditions
B(0, x) = B0(x), E(0, x) = E0(x) for any x ∈ Ω. (13)
In accordance with the basic principles of continuum mechanics, the magnetofluid dynamics is described (see Cabannes [12]) by the Navier-Stokes-Fourier system of equations with magnetic sources. This in-cludes mass conservation
∂t̺ + divx(̺u) = 0, (14)
the momentum balance
and the energy balance ∂t 1 2̺|u| 2+ ̺e+ div x (1 2̺|u|
2+ ̺e + p)u − Su+ div
xq = E · j + g, (16)
where e is the specific internal energy, q denotes the heat flux, T is the Cauchy stress tensor, f is a given body force, g is a given energy source and the Lorentz force ̺cE + j ×B is imposed by the electromagnetic
field. In the magnetohydrodynamic approximation, it may be shown [12] that this approximation also implies that ̺cE is negligible compared to j × B. Therefore, the momentum balance (15) reads
∂t(̺u) + divx(̺u ⊗ u) = divxT+ j × B + f. (17)
Furthermore the Cauchy stress tensor T is given by Stokes’ law
T= S − pI, (18)
where p is the pressure, and the symbol S stands for the viscous stress tensor, given by S= ν ∇xu + ∇xut− 2 3divx(u) I + η divx(u) I, (19)
with the shear viscosity coefficient ν, and the bulk viscosity coefficient η. Finally we impose a no-slip boundary condition for the velocity
u|∂Ω= 0. (20)
Using (11) and (16) the energy of the system satisfies ∂t 1 2̺|u| 2 + ̺e + M(|H|)+ divx 1 2̺|u| 2+ ̺e + p u + E × B − Su + q = 0, (21) where e is the specific internal energy, and q denotes the heat flux.
We prescribe a no-flux boundary condition for q
q · n|∂Ω= 0. (22)
Using the previous boundary conditions we deduce that the (conserved) total energy of the system satisfies d dt Z Ω 1 2̺|u| 2 + ̺e + M(|H|)dx = 0. (23) In the variational formulation it will be more convenient to replace (21) by the entropy balance
∂t(̺s) + divx(̺su) + divx
q ϑ
= r, (24)
with the specific entropy s, and the entropy production rate r, for which the second law of thermodynamics requires r ≥ 0.
The specific entropy s, the specific internal energy e, and the pressure are interrelated through the second principle of thermodynamics
ϑDs = De + D1 ̺
p, (25)
where D stand for the differential with respect to the variable ̺, ϑ, which implies some compatibility conditions between e and p ( Maxwell’s relation).
If the motion is smooth, the sum of the kinetic and magnetic energy satisfies ∂t 1 2̺|u| 2 + M(|H|) + divx 1 2̺|u| 2+ p u + E × B − Su = p divx(u) − S : ∇xu − 1 µσ|curlxB| 2. (26)
Indeed using (25) one checks that r = 1 ϑ S: ∇xu + 1 σ|curlxH| 2 −q · ∇ϑxϑ . (27)
If the motion is not (known to be) smooth, validity of (16) is no longer guaranteed and the dissipation rate of mechanical energy may exceed the value S : ∇xu.
Accordingly, we replace equation (24) by inequality
∂t(̺s) + divx(̺su) + divx
q ϑ ≥ϑ1S: ∇xu + 1 σ|curlxH| 2 −q · ∇ϑxϑ, (28) to be satisfied together with the total energy balance (23).
1.2
Hypotheses on constitutive relations
In order to close the system one needs first to postulate a constitutive equation relating the pressure to the other variables. Several possibilities have been considered. We choose here, in the spirit of [30] (however see [36] [39] [51] for other possible choices) to focus on high-temperature phenomena appearing in various astrophysical contexts (stars, planetar magnetospheres, neighborhood of black holes etc...) [20] [89] [100] [104], or plasma physics applications (tokamaks, inertial confinement fusion devices etc...) [64] [67].
In that case, the so called equilibrium diffusion limit of radiation hydrodynamics involves a modified pressure law where the pressure pF(̺, ϑ) in the fluid is augmented by a radiation component pR(ϑ) related
to the absolute temperature ϑ through the Stefan-Boltzmann law pR(ϑ) =
a 3ϑ
4, with a constant a > 0 (29)
(see, for instance, Chapter 15 in [33]). Accordingly, the specific internal energy of the fluid must be supplemented by a term
eR(ϑ) = eR(̺, ϑ) =
a ̺ϑ
4, (30)
where ̺ is the fluid density and similarly, the heat conductivity of the fluid is enhanced by a radiation component (see [11] [78])
qR(ϑ) = −κRϑ3∇xϑ, with a constant κR> 0 (31)
Finally the total pressure p takes the form
p(̺, ϑ) = pF(̺, ϑ) + pR(ϑ), (32)
where the radiation component is given by (29). Similarly, the specific internal energy reads
e(̺, ϑ) = eF(̺, ϑ) + eR(̺, ϑ), (33)
with eR determined by (30).
Furthermore we suppose that the gas is perfect and monoatomic pF(̺, ϑ) =
2
3̺eF(̺, ϑ). (34)
The following hypothesis expresses the convexity of free energy ∂pF(̺, ϑ)
∂̺ > 0,
∂eF(̺, ϑ)
and finally, we suppose
lim inf
̺→0+ pF(̺, ϑ) = 0, lim inf̺→0+
∂pF(̺, ϑ)
∂̺ > 0 for any fixed ϑ > 0, (36) and
lim inf
ϑ→0+ eF(̺, ϑ) = 0, lim infϑ→0+
∂eF(̺, ϑ)
∂̺ > 0 for any fixed ̺ > 0. (37) As one can check by solving (25) with (34), there exists a function PF ∈ C1(0, ∞) such that
pF(̺, ϑ) = ϑ 5 2P F ̺ ϑ32 . (38) Consequently ∂eF(̺, ϑ) ∂ϑ = 3 2 1 Y 5 3PF(Y ) − P ′ F(Y )Y , with Y = ̺ ϑ32 , so according to (35) PF′(z) > 0, 5 3PF(z) − P ′ F(z)z > 0 for any z > 0, (39)
where the latter inequality yields
PF(z) z53 ′ < 0 for all z > 0. (40) In particular PF(z) z53 → p∞> 0 for z → ∞. (41) According to (36), (37), we see that PF ∈ C1[0, ∞). Moreover
0 < lim inf z→0+ 1 z 5 3PF(z) − P ′ F(z)z ≤ lim sup z→0+ 1 z 5 3PF(z) − P ′ F(z)z < ∞, (42) lim sup z→∞ 1 z 5 3PF(z) − P ′ F(z)z < ∞, (43) and lim z→0+PF(z) = 0, limz→0+P ′ F(z) > 0, lim z→∞ P′ F(z) z23 = 5 3p∞> 0. (44) Now it follows from (44) that
P′
F(z) ≥ cz
2
3 for all z > 0, and a certain c > 0.
Therefore there exists pc> 0 such that the mapping
̺ 7→ pF(̺, ϑ) − pc̺
5
3, (45)
is a non-decreasing function of ̺ for any fixed ϑ > 0. Accordingly, using equation (25) we have
s(̺, ϑ) = sF(̺, ϑ) + sR(̺, ϑ), (46) where sF(̺, ϑ) = SF ̺ ϑ32 , with SF′ (z) = − 3 2 5 3PF(z) − PF′(z)z z2 , (47) and ̺sR(̺, ϑ) = 4 3aϑ 3. (48)
Note that after (39) and (42), SF is a decreasing function such that lim z→0+zS ′ F(z) = − 2 5P ′ F(0) < 0,
whence, normalizing by SF(1) = 0, we get
−c1log(z) ≤ SF(z) ≤ −c2log(z), c1> 0, for all 0 < z ≤ 1, (49)
and
0 ≥ SF(z) ≥ −c3log(z) for all z ≥ 1. (50)
Using the second law of thermodynamics, we see that viscosity coefficients ν and η are non-negative quantities.
In addition, we shall suppose that
ν = ν(ϑ, |H|) > 0, η = η(ϑ, |H|) ≥ 0, (51) where the bulk viscosity ν satisfies some technical, but physically relevant, coercivity conditions to be specified below.
Similarly, the heat flux q obeys Fourier’s law
q = qR+ qF, (52)
where the radiation heat flux qR is given by (31) and
qF = −κF(̺, ϑ, |H|)∇xϑ, κ > 0. (53)
For the sake of simplicity we assume the transport coefficients ν, κF, µ and σ to admit a common
temperature scaling, namely
c1(1 + ϑα) ≤ κF(̺, ϑ, |H|), ν(ϑ, |H|), σ−1(̺, ϑ, |H|) ≤ c2(1 + ϑα), c1, c2> 0, (54)
with α ≥ 1 to be specified below. Similarly, we suppose that
0 ≤ η(ϑ, |H|) ≤ c3(1 + ϑα), (55)
and that the magnetic permeability has the symbol property
cks(1 + s)−k≤ ∂sk(sµ(s)) ≤ cks(1 + s)−k, (56)
for any s ≥ 0 and for k = 0, 1, with ck, ck> 0.
We finally suppose that external sources are absent: f = 0 and g = 0.
Let us conclude this introduction by briefly fixing the limits of our study. As our framework is explicitely viscous and compressible, all the important works about the incompressible model are beyond our scope. However the reader may consult, among standard references: Ladyzhenskaya and Solonnikov [69], Duvaut and Lions [31] [32], Sermange and Temam [88] and Cannone et al. [13] for reviews of the incompressible setting and Germain and Masmoudi [46] (with references therein) for the inviscid case. Finally we refer the reader to Eringen and Maugin [34] for more complex situations coupling electrodynamics to continua.
The article is structured as follows: in Section 2 we consider the main existence result of weak solutions, in Section 3 we present related models (barotropic or polytropic) and in Section 4 we introduce asymptotic models. In Section 5 we present the main results concerning existence and uniqueness of strong solutions including critical aspects and finally results for one dimensional models are given in the last Section 6.
2
Weak solutions
2.1
Variational formulation
As in the pure hydrodynamical model [29] the approach is based on the concept of variational solutions. 1- The mass conservation is expressed through the weak form
Z T 0 Z Ω ̺∂tϕ + ̺u · ∇xϕ dx dt + Z Ω ̺0ϕ(0, ·) dx = 0, (57)
to be satisfied for any test function ϕ ∈ D([0, T ) × R3) where ̺
0 is the initial density.
Assuming that both ̺ and the flux ̺u are locally integrable on [0, T ) × Ω equation (57) makes sense. Moreover if ̺ belongs to L∞(0, T ; Lγ(Ω)) for a γ > 1, then [82] ̺ ∈ C([0, T ]; Lγ
weak(Ω)) i.e.
̺(t) → ̺0 weakly in Lγ(Ω) as t → 0,
and the total mass
M0= Z Ω ̺(t) dx = Z Ω ̺0 dx, (58) is a constant of motion.
For γ ≥ 2 one can use the DiPerna-Lions theory [28] to show that the pair (̺, u) satisfies the renor-malized equation Z T 0 Z Ω
b(̺)∂tϕ + b(̺)u · ∇xϕ + (b(̺) − b′(̺)̺)divx(u)ϕ
dx dt + Z
Ω
b(̺0)ϕ(0, ·) dx = 0, (59)
for any ϕ ∈ D([0, T ) × R3), and for any continuously differentiable function b whose derivative
van-ishes for large arguments. It can also be shown that any renormalized solution ̺ belongs to the space C([0, T ]; L1(Ω)) (see [28]).
2- The variational formulation of the momentum equation reads now Z T
0
Z
Ω
(̺u) · ∂tϕ + (̺u ⊗ u) : ∇xϕ + p divx(ϕ)
dx dt = Z T 0 Z Ω S: ∇xϕ − (j × B) · ϕ dx dt − Z Ω (̺u)0· ϕ(0, ·) dx, (60)
which must be satisfied for any vector field ϕ ∈ D([0, T ) × Ω; R3). Once more we suppose that the
quantities ̺u, ̺u ⊗ u, p, S, ̺∇xΨ, and J × B are at least locally integrable on the set [0, T ) × Ω.
The initial value of the momentum (̺u)0 must satisfy
(̺u)0= 0 a.e. on the set {̺0= 0}. (61)
Finally using the no-slip boundary condition (20) we get
u ∈ L2(0, T ; W01,2(Ω; R3)). (62)
Observe that our choice of the function space has been motivated by the a priori bounds resulting from the entropy balance specified below.
3- According to (28) the weak form of the entropy production reads Z T 0 Z Ω ̺s ∂tϕ + ̺su · ∇xϕ +q ϑ· ∇xϕ dx dt ≤ (63) Z T 0 Z Ω 1 ϑ q · ∇xϑ ϑ − S : ∇xu − 1 µ2σ|curlxB| 2 ϕ dx dt − Z Ω (̺s)0ϕ(0, ·) dx,
for any test function ϕ ∈ D([0, T ) × R3), ϕ ≥ 0.
We observe that the presence of ϑ in the denominator forces this quantity to be positive on a set of full measure for (63) to make sense.
4- Finally Maxwell’s equation (9) is replaced by Z T 0 Z Ω B · ∂tϕ − B × u +σ1curlx B µ · curlxϕ dx dt + Z Ω B0· ϕ(0, ·) dx, (64)
to be satisfied for any vector field ϕ ∈ D([0, T ) × R3).
Hereafter using boundary conditions (12) and (62) one must take
B0∈ L2(Ω), divx(B0) = 0 in D′(Ω), B0· n|∂Ω= 0. (65)
By virtue of Theorem 1.4 in [93], B0 belongs to the closure of solenoidal fields on D(Ω) with respect to
the L2−norm.
Applying (23) and (28), we infer that M(|H|) ∈ L∞(0, T ; L1(Ω; R3)) and curl
xH ∈ L2(0, T ; L2(Ω; R3)).
But using (10) and (56) we deduce that B ∈ L∞(0, T ; L2(Ω; R3)), and from (4) and (12) we see that
divx(B)(t) = 0 in D′(Ω), B(t) · n|∂Ω= 0 for a.e. t ∈ (0, T ),
so using Theorem 6.1 in [32] we conclude that
H ∈ L2(0, T ; W1,2(Ω; R3)), divx(B)(t) = 0, B · n|∂Ω= 0 for a.e. t ∈ (0, T ). (66)
5- According to (21) we shall assume the total energy to be a constant of motion
E(t) = Z Ω 1 2̺(t)|u(t)| 2
+ (̺e)(t) + M(|H(t)|)dx = E0 for a.e. t ∈ (0, T ), (67)
where E0= Z Ω 1 2̺0|(̺u)0| 2+ (̺e) 0+ M(|H0|) dx. (68)
Of course the initial values of the entropy (̺s)0and the internal energy (̺e)0should be chosen consistently
with the constitutive relations. A natural possibility consists in fixing the initial temperature ϑ0 and
setting
(̺s)0= ̺0s(̺0, ϑ0), (̺e)0= ̺0e(̺0, ϑ0). (69)
2.2
Global existence of weak solutions for large data
The following result holds
Theorem 2.1 Let Ω ⊂ R3 be a bounded domain with boundary of class C2+δ, δ > 0.
Suppose that the thermodynamic functions
p = p(̺, ϑ), e = e(̺, ϑ), s = s(̺, ϑ)
are interrelated through (25), where p, e can be decomposed as in (32), (33), with the components pF,
eF satisfying (34). Moreover, let pF(̺, ϑ), eF(̺, ϑ) be continuously differentiable functions of positive
arguments ̺, ϑ satisfying (35)-(37).
Furthermore, we suppose that the transport coefficients
are continuously differentiable functions of their arguments obeying (51)-(56), with
1 ≤ α <6527. (70) Finally, let the initial data ̺0, (̺u)0, ϑ0, B0 be given so that
̺0∈ L 5 3(Ω), (̺u) 0∈ L1(Ω; R3), ϑ0∈ L∞(R3), B0∈ L2(Ω; R3), (71) ̺0≥ 0, ϑ0> 0, (72) (̺s)0= ̺0s(̺0, ϑ0), 1 ̺0|(̺u) 0|2, (̺e)0= ̺0e(̺0, ϑ0) ∈ L1(Ω), (73) and divx(B0) = 0 in D′(Ω), B0· n|∂Ω= 0. (74)
Then problem (57)-(65) possesses at least one variational solution ̺, u, ϑ, B in the sense of Section 2.1 on an arbitrary time interval (0, T ).
Observe that hypothesis (70) is a technical restriction. However functional dependence on ϑ was rigorously justified in [26] after an asymptotic analysis of some kinetic-fluid models. The same temperature scaling on all transport coefficients was also suggested in [47].
2.2.1 Difficulties and methods
The main difficulty of the problem being the lack of a priori estimates, methods of weak convergence based on the theory of the parametrized (Young) measures represent the main tool [83] to proceed. Accordingly, we denote by b(z) a weak limit of any sequence of composed functions {b(zn)}n≥1. More precisely,
Z RM b(y, z)ϕ(y) dy = Z RM ϕ(y) Z RK b(y, z) dΛy(z) dy,
where Λy(z) is a parametrized measure associated to a sequence {zn}n≥1 of vector-valued functions,
zn: RM → RK (see Chapter 1 in [83]).
Assume there is a sequence of approximate solutions resulting from a suitable regularization process. The starting point is to establish the relation
p(̺, ϑ) −43ν + ηdivx(u) b(̺) = p(̺, ϑ) b(̺) −43ν(ϑ, |B|) + η(ϑ, |B|)divx(u) b(̺), (75)
for any bounded function b.
The quantity p−(4/3ν +η)divx(u) is called the effective viscous pressure and relation (75) was proved
by P.-L. Lions [72] for the barotropic Navier-Stokes system, with p = p(̺), when viscosity coefficients ν and η are constant. The same result for general temperature dependent viscosity coefficients was obtained in [40] with the help of certain commutator estimates in the spirit of [19].
In the present setting, this approach has to be modified in order to accommodate the dependence of ν and η on the magnetic field B.
The propagation of density oscillations can be described by the renormalized continuity equation [28] ∂tb(̺) + divx(b(̺)u) +
b′(̺)̺ − b(̺)divx(u) = 0, (76)
and its “weak” counterpart
∂tb(̺) + divx(b(̺)u) +
To prove (76) one must show first boundedness of the oscillations defect measure oscγ+1[̺n→ ̺](Q) = sup k≥1 lim sup n→∞ Z Q|T k(̺n) − Tk(̺)|γ+1dx dt < ∞, (78)
where Tk(̺) = min{̺, k}, for any bounded Q ⊂ R4 and a γ > 1.
Due to the poor estimates resulting from (54), (55) and (63) relation (78) has to be replaced by a weaker weighted estimate
sup k≥1 lim sup n→∞ Z Q (1 + ϑn)−β|Tk(̺n) − Tk(̺)|γ+1 dx dt < ∞, (79) for suitable β > 0, γ > 1.
Finally one recovers strong convergence of the sequence {ϑn}n≥1of approximate temperatures,
know-ing that spatial gradients of ϑn are uniformly square integrable and that
̺s(̺, ϑ)ϑ = ̺s(̺, ϑ)ϑ, (80) where (80) can be deduced from the entropy inequality (63) by using Aubin-Lions lemma.
The structure of the proof of Theorem 2.1 goes as follows: introducing a three level approximation scheme adapted from [39] allows to reduce the proof to a weak stability problem. One first proves uniform bounds on the sequence of approximate solutions. Relying on these uniform bounds, one can handle the convective terms in the field equations by means of a compactness argument. The strong convergence of the sequence of approximate temperatures is proved by using entropy inequality together with the theory of parametrized (Young) measures discussed below. Then one finally shows that the sequence of approximate densities converges strongly in the Lebesgue space L1((0, T ) × Ω). This requires
weighted estimates of the oscillations defect measure in order to show that the limit densities satisfy the renormalized continuity equation.
2.2.2 The approximation scheme
The approximation scheme used in the following is inspired from Chapter 3 in [42]. 1- The continuity equation (17) is regularized by a viscous approximation
∂t̺ + divx(̺u) = ε∆̺, ε > 0, (81)
on (0, T ) × Ω, supplemented by the homogeneous Neumann boundary conditions
∇x̺ · n|∂Ω= 0. (82)
The initial density is given by
̺(0, ·) = ̺0,δ, (83)
where
̺0,δ ∈ C1(Ω), ∇x̺0,δ· n|∂Ω= 0, inf
x∈Ω̺0,δ(x) > 0, (84)
with a positive parameter δ > 0.
The functions ̺0,δ are chosen in such a way that
̺0,δ → ̺0 in L
5
3(Ω), |{̺
0,δ < ̺0}| → 0 for δ → 0. (85)
Here, of course, the choice of the “critical” exponent γ = 5/3 is directly related to estimate (44) established above.
2- The regularized momentum equation is
∂t(̺u) + divx(̺u ⊗ u) + ∇xp + δ∇x̺Γ+ ε∇xu∇x̺ = divx(S) + j × B, (86)
in (0, T ) × Ω where the approximate velocity field satisfies the no slip boundary condition
u|∂Ω= 0. (87)
We also prescribe initial conditions
(̺u)(0, ·) = (̺u)0,δ, (88) where (̺u)0,δ= (̺u)0 provided ̺0,δ≥ ̺0, 0 otherwise. (89)
The artificial pressure term δ̺Γ in (86) provides additional estimates on the approximate densities in
order to facilitate the limit passage ε → 0. To this end, one has to take Γ large enough, say, Γ > 8, and to re-parametrize the initial distribution of the approximate densities so that
δ Z
Ω
̺Γ
0,δ dx → 0 for δ → 0. (90)
3- The entropy equation (24) is replaced by the regularized internal energy balance ∂t(̺e + δ̺ϑ) + divx
(̺e + δ̺ϑ)u− divx
(κF+ κRϑ3+ δϑΓ)∇xϑ (91) = S : ∇xu − p divx(u) + 1 σ|curlxH| 2 + εΓ|∇x̺|2̺Γ−2,
to be satisfied in (0, T ) × Ω, together with no-flux boundary conditions
∇xϑ · n|∂ = 0. (92)
The initial conditions read
̺(e + δϑ)(0, ·) = ̺0,δ(e(̺0,δ, ϑ0,δ) + δϑ0,δ), (93)
where the approximate temperature satisfies
ϑ0,δ∈ C1(Ω), ∇xϑ0,δ· n|∂Ω= 0, inf x∈Ωϑ0,δ(x) > 0, (94) and ϑ0,δ→ ϑ0in Lp(Ω) for any p ≥ 1, δR Ω̺0,δlog(ϑ0,δ) dx → 0, R Ω̺0,δs(̺0,δ, ϑ0,δ) dx → R Ω̺0s(̺0, ϑ0) dx, (95) as δ → 0, with Z Ω
̺0,δe(̺0,δ, ϑ0,δ) dx < c uniformly for δ > 0. (96)
Finally, the magnetic induction vector B obeys the original Maxwell’s equations
∂tB + curlx(B × u) + curlx 1 σcurlx B µ = 0, divxB = 0 (97)
in (0, T ) × Ω, supplemented with the initial condition
B(0, ·) = B0,δ, (98)
where, by virtue of Theorem 1.4 in [93], one can take
B0,δ ∈ D(Ω; R3), divxB0,δ= 0, (99)
B0,δ→ B0 in L2(Ω; R3) for δ → 0. (100)
For given positive parameters ε, δ , and Γ > 8, the proof of Theorem 2.1 consists in the following steps: Step 1 Solving problem (81)-(100) for fixed ε > 0, δ > 0.
Step 2 Passing to the limit for ε → 0. Step 3 Letting δ → 0.
One sees that Step 1 can be achieved by using a simple fixed point argument: equation (86) is solved in terms of the velocity u with the help of the Faedo-Galerkin method, where ̺, ϑ, and B are computed successively from (81), (91), and (97) as functions of u. It is easy to check that the corresponding approximate solutions satisfy the energy balance
d dt Z Ω 1 2̺|u| 2 + ̺e + M(|H|) + δ Γ − 1̺ Γ+ δ̺ϑdx = 0, (101)
in D′(0, T ), and the limit property
lim t→0+ Z Ω 1 2̺|u| 2 + ̺e + M(|H|) + δ Γ − 1̺ Γ+ δ̺ϑdx = (102) Z Ω 1 2̺0,δ|(̺u)0,δ| 2+ ̺ 0,δe(̺0,δ, ϑ0,δ) + M(|H0,δ|) + δ Γ − 1̺ Γ 0,δ+ δ̺0,δϑ0,δ dx.
The reason for splitting Steps 2 and 3 into the ε and δ−parts is a refined density estimates based on multipliers ∇x(−∆)−1[̺β], where one must take β = 1 when the artificial viscosity term ε∆̺ is present,
while uniform estimates require β to be a small positive number. For this reason, we focus only on Step 3: the goal is to establish the weak sequential stability property for the solution set of the approximate problem:
1. The density ̺δ ≥ 0 and the velocity uδ satisfy
Z T 0 Z Ω ̺δ∂tϕ + ̺δuδ· ∇xϕ dx dt + Z R3 ̺0,δϕ(0, ·) dx = 0 (103)
for any test function ϕ ∈ D([0, T ) × R3). In addition,
uδ(t) ∈ W01,2(Ω; R3) for a.e. t ∈ (0, T ). (104)
2. The momentum equation Z T 0 Z Ω ̺δuδ· ∂tϕ + (̺δuδ⊗ uδ) : ∇xϕ + pδ divx(ϕ) + δ̺Γδ divx(ϕ) dx dt = (105) Z T 0 Z Ω Sδ : ∇xϕ − (jδ× Bδ) · ϕ dx dt − Z Ω (̺u)0,δ· ϕ(0, ·) dx
holds for any ϕ ∈ D([0, T ) × Ω; R3), where p
δ = p(̺δ, ϑδ), and Sδ, Jδ are determined in terms of uδ, ϑδ,
3. The entropy production inequality reads now Z T 0 Z Ω (̺δsδ+ δ̺δlog(ϑδ))∂tϕ + (̺δsδuδ+ δ̺δlog(ϑδ)uδ) · ∇xϕ dx dt (106) + Z T 0 Z Ω qδ ϑδ · ∇x ϕδ− δϑΓ−1δ ∇xϑδ· ∇xϕ dx dt ≤ Z T 0 Z Ω 1 ϑδ qδ· ∇xϑδ ϑδ − Sδ : ∇x uδ− 1 σδ |curlx Hδ|2− δϑΓ−1δ |∇xϑδ|2 ϕ dx dt − Z Ω ̺0,δs(̺0,δ, ϑ0,δ) + δ̺0,δlog(ϑ0,δ) ϕ(0, ·) dx, for any ϕ ∈ D([0, T ) × R3), ϕ ≥ 0. Here, ϑ
δ is assumed to be positive a.e. on the set (0, T ) × Ω,
sδ = s(̺δ, ϑδ), σδ = σ(̺δ, ϑδ), and qδ is a function of ̺δ, ϑδ given by (52).
4. The magnetic induction vector Bδ satisfies
Z T 0 Z Ω Bδ· ∂tϕ − (Bδ× uδ+ 1 σδ curlxHδ) · curlxϕ dx dt + Z Ωδ B0,δ· ϕ(0, ·) dx = 0, (107)
for any vector field ϕ ∈ D([0, T ) × R3; R3).
5. Finally, the (total) energy equality Z Ω 1 2̺δ|uδ| 2+ ̺ δe + M(Hδ) + δ Γ − 1̺ Γ δ + δ̺δϑδ (t) dx (108) = Z Ω 1 2̺0,δ|(̺u)0,δ| 2+ ̺ 0,δe(̺0,δ, ϑ0,δ) + M(H0,δ) dx + Z Ωδ δ Γ − 1̺ Γ 0,δ+ δ̺0,δϑ0,δ dx, holds for a.e. t ∈ (0, T ).
The weak sequential stability problem to be addressed below consists now in showing that one can pass to the limit
̺δ→ ̺, uδ → u, ϑδ → ϑ, Bδ→ B,
as δ → 0, in a suitable topology, where the limit quantity {̺, u, ϑ, B} is a variational solution of problem (57)-69.
2.2.3 Uniform bounds
One must identify uniform bounds on the sequences {̺δ}δ>0, {uδ}δ>0, {ϑδ}δ>0, and {Bδ}δ>0 through
total energy balance (108), dissipation inequality (106) and the other relations resulting from (103)-(108). 1. Total mass conservation. As ̺δ, ̺δuδ are locally integrable in [0, T ) × Ω, it follows from (103) that
total mass is a constant of motion, specifically, Z Ω ̺δ(t) dx = Z Ω ̺0,δ dx = M0 for a.e. t ∈ (0, T ). (109)
In particular, as ̺δ ≥ 0 and (85) holds, we get
{̺δ}δ>0 bounded in L∞(0, T ; L
1(Ω)). (110)
2. Energy estimates. Using (36) and (44) one gets Z Ω ̺δe(̺δ, ϑδ) dx = 3 2 Z Ω pF(̺δ, ϑδ) dx ≥ 3p∞ 2 Z Ω ̺ 5 3 δ dx.
Total energy balance (108) together with bounds on the initial data (85), (89), (90), (96), and (100) yield the energy estimates
{̺δ}δ>0 bounded in L∞(0, T ; L 5 3(Ω)), (111) {ϑδ}δ>0 bounded in L∞(0, T ; L4(Ω)), (112) {Hδ}δ>0 bounded in L∞(0, T ; L 2(Ω; R3)), (113) ̺δ|uδ|2 δ>0, {̺δe(̺δ, ϑδ)}δ>0 bounded in L∞(0, T ; L1(Ω)), (114) and δ Z Ω
̺Γδ dx ≤ c uniformly with respect to δ > 0. (115)
In particular, using H¨older’s inequality, (111) and (114) imply {(̺u)δ}δ>0 bounded in L∞(0, T ; L
5
4(Ω; R3)). (116)
3. Dissipation estimates. Taking a spatially homogeneous test function ϕ such that ϕ(0, ·) = 1 in the entropy production inequality (106) and using (95) (96), we obtain
Z τ 0 Z Ω 1 ϑδ Sδ: ∇xuδ− qδ· ∇xϑδ ϑδ + 1 σδ|curlx Hδ|2+ δϑΓ−1δ |∇xϑδ|2 dx dt (117) ≤ c1+ Z Ω ̺δ(τ )s(̺δ(τ ), ϑδ(τ )) + δ̺δ(τ ) log(ϑδ(τ )) dx ≤ c2+ Z Ω ̺δ(τ )s(̺δ(τ ), ϑδ(τ )) dx,
for a.e. τ ∈ (0, T ), where the last inequality follows from (111), (112). Using (49) and (50), the integral in the right hand side may be estimated
Z Ω ̺δs(̺δ, ϑδ) dx = Z Ω 4 3aϑ 3 δ+ ̺δSF ̺δ ϑ32 δ ! dx ≤ c1− c2 Z {̺δ≤ϑ 3 2 δ} ̺δ log(̺δ) − 3 2log(ϑδ) dx ≤ c3+ c4 Z {̺δ≤ϑ 3 2 δ} ̺δlog(ϑδ) dx ≤ c5+ c6 Z Ω ̺δϑδ dx ≤ c7, (118)
where we have taken into account (111) and (112).
Thus the integral in the left hand side of (117) is bounded independently of δ and we conclude after (53)-(56) that n (1 + ϑδ) α−1 2 h∇xuδi o δ>0 is bounded in L 2(0, T ; L2(Ω; R3×3)), (119) n (1 + ϑδ) α−1 2 divx(uδ) o δ>0 is bounded in L 2(0, T ; L2(Ω)), (120) {∇xlog(ϑδ)}δ>0, n ∇xϑ 3 2 δ o δ>0 are bounded in L 2(0, T ; L2(Ω; R3)), (121) n (1 + ϑδ) α−1 2 curl xHδ o δ>0 is bounded in L 2(0, T ; L2(Ω; R3)), (122) and n√ δ∇xϑ Γ 2 δ o δ>0 is bounded in L 2(0, T ; L2(Ω; R3)), (123)
where α satisfies (70), and where we denote by
hQi = 12(Q + QT) −13trace(Q)I, the traceless component of the symmetric part of a tensor Q.
Since the velocity field satisfies (104) we find {uδ}δ>0 bounded in L
2(0, T ; W1,2
0 (Ω; R3)), (124)
and (121) and (112) lead to n ϑ32 δ o δ>0 bounded in L 2(0, T ; W1,2(Ω)). (125)
4. Positivity of the absolute temperature. According to physics, temperature must be positive a.e. on (0, T ) × Ω. To justify that we use the uniform L2− estimates of ∇
xlog(ϑδ) given by (124) and the
following version of Poincar´e’s inequality:
Lemma 2.1 Let Ω ⊂ RN be a bounded Lipschitz domain, and ω ≥ 1 be a given constant. Furthermore,
assume O ⊂ Ω is a measurable set such that |O| ≥ o > 0. Then
kvkW1,2(Ω)≤ c(o, Ω, ω) k∇xvkL2(Ω;R3)+ ( Z O|v| 1 ω dx)ω .
In order to apply Lemma 2.1, we show first the uniform bound
0 < T ≤ Z Ω ϑδ(t) dx for a.e. t ∈ (0, T ). (126) As a consequence of (106) we get Z Ω ̺δs(̺δ, ϑδ) + δ̺δlog(ϑδ) (t) dx ≥ Z Ω ̺0,δs(̺0,δ, ϑ0,δ) + δ̺0,δlog(ϑ0,δ) dx,
where, by virtue of (85) and (95)
δ Z Ω ̺0,δlog(ϑ0,δ) dx → 0 and Z Ω ̺0,δ s(̺0,δ, ϑ0,δ) dx → Z Ω ̺0s(̺0, ϑ0) dx.
On the other hand (111) and (112) yield
δ Z Ω ̺δlog(ϑδ)(t) dx ≤ δ Z Ω ̺δ(t)ϑδ(t) dx → 0 for a.e t ∈ (0, T ), while, using (46) Z Ω ̺δs(̺δ, ϑδ)(t) dx = Z Ω 4 3aϑ 3 δ+ ̺δSF ̺δ ϑ32 δ !! (t) dx.
Supposing the opposite of (126), one could extract a sequence {ϑδ(tδ)}δ>0 such that
ϑδ(tδ) → 0 weakly in L4(Ω), and strongly in Lp(Ω) for any 1 ≤ p < 4, (127)
Z Ω 4 3aϑ 3 0+ ̺0SF ̺0 ϑ32 0 !! dx ≤ lim inf δ→0+ Z Ω ̺δ(tδ)SF ̺δ ϑ32 δ (tδ) dx, (128) where ̺δ(tδ) → ̺(t) weakly in L 5 3(Ω), Z Ω ̺δ(tδ) dx = Z Ω ̺0,δ dx.
Now, for any fixed K > 1, one can write Z Ω ̺δ(tδ)SF ̺δ ϑ32 δ (tδ) dx (129) ≤ Z {̺δ(tδ)≤Kϑ 3 2 δ(tδ)} ̺δ(tδ)SF ̺δ ϑ32 δ ! (tδ) dx + Z {̺δ(tδ)≥Kϑ 3 2 δ(tδ)} ̺δ(tδ)SF ̺δ ϑ32 δ ! (tδ) dx ≤ c Z {̺δ(tδ)≤Kϑ 3 2 δ(tδ)} ̺δ(tδ) 1 + ϑ32 δ ̺δ (tδ) ! dx + SF(K) Z Ω ̺0,δ dx ≤ 2c Z Ω ϑ32 δ(tδ) dx + SF(K) Z Ω ̺0,δ dx.
Thus combining (127)-(129) one concludes Z Ω 4 3aϑ 3 0+ ̺0SF ̺0 ϑ32 0 dx ≤ SF(K) Z Ω ̺0dx for any K > 1,
which is in contradiction with the fact that SF is strictly decreasing with limK→∞SF(K) < 0, and ϑ0
positive (non-zero) on Ω. This proves the uniform bound (126). Finally, observing that
T − ε|Ω| ≤ Z {ϑδ(t)>ε} ϑδ(t) dx ≤ |{ϑδ(t) > ε}| 3 4kϑ δ(t)kL4(Ω) for any ε > 0,
we deduce, using (112), that there exist ε > 0 and o such that
|{ϑδ(t) > ε}| > o > 0 for a.e. t ∈ (0, T ),
uniformly for δ > 0. Then using (121) one can apply Lemma 2.1 to log(ϑδ) and we obtain the required
estimate
{log(ϑδ)}δ>0 bounded in L
2(0, T ; W1,2(Ω)). (130)
5. Estimates of the magnetic field. Using the integral identity (107) we get
divx(Bδ)(t) = 0 in D′(Ω), Bδ· n|∂Ω= 0, (131)
which, together with estimates (113), (122) and Theorem 6.1 in [32], yields
{Hδ}δ>0 bounded in L2(0, T ; W1,2(Ω; R3)). (132)
6. Pressure estimates. We first observe that estimates (111), (124) imply boundedness of the sequences {̺δuδ}δ>0and {̺δuδ⊗ uδ}δ>0 in Lp((0, T ) × Ω) for a certain p > 1.
Now one gets
Sδ =pϑδν(ϑδ) s ν(ϑδ) ϑδ h∇ xuδi +pϑδη(ϑδ) s η(ϑδ) ϑδ divx(uδ), (133)
where, by virtue of (112), (119), (125), and hypothesis (70), the expression on the right-hand side is bounded in Lp((0, T ) × Ω) for a certain p > 1.
Finally the Lorentz force jδ× Bδ = curlxHδ× µ(Hδ)Hδ is controlled by (113), (132) and (56) which,
combined with a simple interpolation argument, yields
At this stage, using the main result of [44], we are allowed to use the quantities
ϕ(t, x) = ψ(t)B[̺ωδ], ψ ∈ D(0, T ) for a sufficiently small parameter ω > 0,
as test functions in (105), where B is the Bogovskii operator [9] solving the boundary value problem
divx B[v]= v − 1 |Ω| Z Ω v dx, B|∂Ω= 0. (135)
The resulting estimate reads Z T 0 Z Ω p(̺δ, ϑδ)̺ωδ + δ̺Γ+ωδ dx dt < c, with c independent of δ, (136) then
{p(̺δ, ϑδ)}δ>0 is bounded in Lp((0, T ) × Ω) for a certain p > 1, (137)
and n ̺53+ω δ o δ>0 is bounded in L 1((0, T ) × Ω). (138)
2.2.4 Sequential stability of the equations
1. Continuity equation. With the estimates established in the previous section, we can pass to the limit δ → 0 in (103).
Indeed, extracting subsequences if necessary, we deduce from (111),(116), (124), and the fact that ̺δ
and uδ satisfy the identity (103)
̺δ → ̺ in C([0, T ]; L 5 3 weak(Ω)), (139) uδ → u weakly in L2(0, T ; W01,2(Ω; R 3)), (140) ̺δuδ→ ̺u weakly-(*) in L∞(0, T ; L 5 4(Ω; R3)), (141)
where the limit quantities satisfy (57).
2.Momentum equation. Using the estimates obtained in Section 2.2.3.7 together with (105) we have ̺δuδ → ̺u in C([0, T ]; L 5 4(Ω; R3)), (142) ̺δuδ⊗ uδ → ̺u ⊗ u weakly in L2(0, T ; L 30 29(Ω; R3×3 sym)), (143)
where we have used the embedding W01,2(Ω) ֒→ L6(Ω).
Furthermore, in accordance with (136), (137)
p(̺δ, ϑδ) → p(̺, ϑ) weakly in Lp((0, T ) × Ω), (144)
and
δ̺Γ
δ → 0 in Lp((0, T ) × Ω), (145)
for a certain p > 1. Here, in agreement with Section 2.2.3, Z T 0 Z Ω p(̺, ϑ)ϕ dx dt = Z T 0 Z Ω ϕ Z R2 p(̺, ϑ) dΛt,x(̺, ϑ) dx dt, ϕ ∈ D((0, T ) × Ω),
where Λt,x(̺, ϑ) is a parametrized (Young) measure associated to the (vector valued) sequence {̺δ, ϑδ}δ>0.
Similarly, one can use (107) together with estimates (113), (132) to deduce
and
µ(Hδ) → µ(H) strongly in L2((0, T ) × Ω; R3),
then
jδ× Bδ → j × B weakly in Lp((0, T ) × Ω; R3),
for a certain p > 1. Thus the limit quantities satisfy an averaged momentum equation Z T
0
Z
Ω
(̺u) · ∂tϕ + (̺u ⊗ u) : ∇xϕ + p(̺, ϑ) divx(ϕ)
dx dt (147) = Z T 0 Z Ω S: ∇xϕ − (j × B) · ϕ dx dt − Z Ω (̺u)0· ϕ(0, ·) dx
for any vector field ϕ ∈ D([0, T ) × Ω; R3).
The symbol S denotes a weak limit in Lp(0, T ; Lp(Ω; R3×3
sym)), p > 1 of the approximate viscosity tensors
Sδspecified in (133). Clearly, relation (147) will coincide with the (variational) momentum equation (60)
as soon as we show strong (pointwise) convergence of the sequences {̺δ}δ>0 and {ϑδ}δ>0.
2.2.5 Entropy inequality and strong convergence of the temperature
1. Entropy inequality. In order to obtain information on the time oscillations of the sequence {ϑδ}δ>0,
we use estimates on ∂t(̺δs(̺δ, ϑδ)) provided by the approximate entropy balance (106).
It first follows from (49), (50) that |̺δs(̺δ, ϑδ)| ≤ c
ϑ3δ+ ̺δ| log(̺δ)| + ̺δ| log(ϑδ)|
,
therefore, by virtue of the uniform estimates (111), (112), (116), and (130), we can assume
̺δs(̺δ, ϑδ) → ̺s(̺, ϑ) weakly in Lp((0, T ) × Ω), (148)
̺δs(̺δ, ϑδ)uδ → ̺s(̺, ϑ)u weakly in Lp((0, T ) × Ω; R3), (149)
for a certain p > 1.
Similarly, one estimates the entropy flux q ϑδ + δϑΓ−1δ ∇xϑδ ≤ c |∇xlog(ϑδ)| + ϑ2δ|∇xϑδ| + δϑΓ−1δ |∇xϑδ| , (150) where ϑ2δ|∇xϑδ| = 2 3ϑ 3 2 δ ∇xϑ 3 2 δ , δϑ Γ−1 δ ∇xϑδ = 2 Γ √ δ ∇xϑ Γ 2 δ √ δϑs Γ 2 δ ϑ (1−s)Γ 2 δ .
Choosing the parameter 0 < s < 1 small enough so that (1 − s)Γ
2 = 1, we have after H¨older’s inequality
δϑΓ−1δ ∇xϑδ Lp(Ω;R3)≤ c √ δ∇xϑ Γ 2 δ L2(Ω;R3)kϑδkL4(Ω) √ δϑs Γ 2 δ L6(Ω).
Then using estimates (112), (123) together with the imbedding W1,2(Ω) ֒→ L6(Ω), we get
Moreover using similar arguments one can also show that q
ϑδ
δ>0
is bounded in Lp((0, T ) × Ω; R3), for a certain p > 1. (152)
On the other hand according to (121)
b(ϑδ) → b(ϑ) weakly in Lq((0, T ) × Ω)), and weakly in L2(0, T ; W1,2(Ω)), (153)
for any finite q ≥ 1 and any function b provided both b and b′ are uniformly bounded.
Now, as a consequence of the entropy balance (106), we have Divt,x
h
̺δs(̺δ, ϑδ) + δ̺δlog(ϑδ), (̺δs(̺δ, ϑδ) + δ̺δlog(ϑδ))uδ+
q ϑδ − δϑ Γ−1 δ ∇xϑδ i ≥ 0 in D′((0, T ) × Ω), while (153) yields Curlt,x h b(ϑδ), 0, 0, 0 i bounded in L2((0, T ) × Ω; R4×4). Thus a direct application of the Div-Curl lemma [80, 92] gives the relation
̺s(̺, ϑ)b(ϑ) = ̺s(̺, ϑ) b(ϑ), (154) for any b as in (153).
Moreover, seeing that the sequence {̺δs(̺δ, ϑδ)ϑδ}δ>0 is bounded in Lp((0, T ) × Ω) we deduce from
(155) the formula
̺s(̺, ϑ)ϑ = ̺s(̺, ϑ)ϑ. (155) 2. Parametrized measures and pointwise convergence of the temperature
To show that (155) implies strong convergence of the sequence {ϑδ}δ>0, we first remark that the
approximate solutions solve the renormalized continuity equation ∂tb(̺δ) + divx(b(̺δ)uδ) +
b′(̺δ)̺δ− b(̺δ)
divx(uδ) = 0 in D′((0, T ) × R3), (156)
provided that ̺δ and uδ are extended by zero outside Ω, and for any continuously differentiable function
b whose derivative vanishes for large arguments.
Functions ̺δ being square-integrable because of the artificial pressure in the energy equality, (156)
follows from (103) via the technique of DiPerna and Lions [28]. Now it follows from (156) that
b(̺δ) → b(̺) in C([0, T ]; Lpweak(Ω)) for any finite p > 1 and bounded b. (157)
Relations (157) and (153) yield
g(̺)h(ϑ) = g(̺) h(ϑ),
or, in terms of the corresponding parametrized measures (see Section 2.2.1)
Λt,x(̺, ϑ) = Λt,x(̺) ⊗ Λt,x(ϑ). (158)
Relation (158) means that oscillations in the sequences {̺δ}δ>0 and {ϑδ}δ>0 are orthogonal, in the
sense that the parametrized measure associated to {̺δ, ϑδ}δ>0 can be written as a tensor product of the
parametrized measures generated by {̺δ}δ>0 and {ϑδ}δ>0.
Let us consider a function H = H(t, x, r, z) defined for t ∈ (0, T ), x ∈ Ω, and (r, z) ∈ R2 through
formula
Clearly, H is a Caratheodory function: H(t, x, ·, ·) is continuous for a.e (t, x) ∈ (0, T ) × Ω, and H(·, ·, r, z) is measurable for any (r, z) ∈ R2.
Moreover, as both sR and sF are increasing functions of the absolute temperature, we have
H(t, x, r, z) ≥ 4 3a
z3− ϑ3(t, x)z − ϑ(t, x)≥ 0. (159) At this stage, we use Theorem 6.2 in [83], namely: weak limits of Caratheodory functions can be expressed in terms of the parametrized measure. Accordingly, we obtain
lim δ→0 Z T 0 Z Ω ϕ(t, x)H(t, x, ̺δ, ϑδ) dx dt = Z T 0 Z Ω ϕ(t, x) Z R2 H(t, x, ̺, ϑ) dΛt,x(̺, ϑ) dx dt = Z T 0 Z Ω ϕ(t, x) Z R2̺s(̺, ϑ)(ϑ − ϑ(t, x))dΛ t,x(̺, ϑ) dx dt − Z T 0 Z Ω ϕ(t, x) Z R2̺s(̺, ϑ(t, x))(ϑ − ϑ(t, x))dΛt,x (̺, ϑ)dx dt = Z T 0 Z Ω ϕ(t, x) Z R2̺s(̺, ϑ)(ϑ − ϑ(t, x))dΛ t,x(̺, ϑ) dx dt = Z T 0 Z Ω ϕ̺s(̺, ϑ)ϑ − ̺s(̺, ϑ)ϑdx dt = 0 for any ϕ ∈ D((0, T ) × Ω, where we have used (155) to get the last equality together with (158), observing that
Z R2̺s(̺, ϑ(t, x))(ϑ − ϑ(t, x)) dΛt,x (̺, ϑ) = Z R ̺s(̺, ϑ(t, x)) dΛt,x(̺) Z R(ϑ − ϑ(t, x)) dΛ t,x(ϑ) = 0.
In particular, we deduce from (159) that ϑ3ϑ = ϑ3ϑ, which is equivalent to
ϑδ → ϑ in L4((0, T ) × Ω). (160)
2.2.6 Pointwise convergence of densities
1. The effective viscous pressure. Let us start with a result of P.-L. Lions [72] on the effective viscous pressure.
In the present setting, it can be stated in terms of the parametrized measures as follows
ψp(̺, ϑ)b(̺) − ψp(̺, ϑ) b(̺) = R : [ψ S]b(̺) − R : [ψ S] b(̺), (161) for any ψ ∈ D(Ω), and any bounded continuous function b, where R = Ri,j is the Riesz operator defined
by means of the Fourier transform:
Ri,j[v] = ∂xi∆−1x ∂xjv = F −1 ξ→x hξiξj |ξ|2Fx→ξ[v] i . (162)
Note that (161) does not depend on the specific form of the constitutive relations for p, S and requires only validity of the momentum equation (105) and the renormalized continuity equation (156) for ̺δ and
uδ.
2. Commutator estimates. If viscosity coefficients ν and η were constant, we would have R[S] = (43ν + η) divx(u), and (161) would give immediately
p(̺, ϑ)b(̺) − p(̺, ϑ) b(̺) = 43ν + η divx(u) b(̺) − 4 3ν + η divx(u) b(̺), (163)
where the quantity p − (4
3ν + η)divx(u) is called the effective viscous pressure.
In order to establish the same relation for variable viscosity coefficients, we write
R : [ψS] = ψ 43ν + η divx(u) + n R : [ψS] − ψ(43ν + η)divx(u) o .
Applying Lemma 4.2 in [40], the quantity
R :hψν(ϑδ, Bδ) h∇xuδi + η(ϑδ, Bδ)divx(uδ)I
i
− ψ(43ν(ϑδ, Bδ) + η(ϑδ, Bδ))divx(uδ),
is bounded in the space L2(0, T ; Wω,p(Ω)) for suitable 0 < ω < 1, p > 1 in terms of the bounds established
in (124), (125) and (132) provided that the viscosity coefficients are globally Lipschitz functions of their arguments.
If this is the case, one can use (157) in order to deduce the desired relation
R : [ψ S]b(̺) − R : [ψ S] b(̺) = ψ(4 3ν + η)divx(u) b(̺) − ψ( 4 3ν + η)divx(u) b(̺) = ψ(4 3ν + η)divx(u) b(̺) − ψ( 4 3ν + η)divx(u) b(̺),
where the last equality is a direct consequence of the pointwise convergence proved in (146) and (160). If ν and η are only continuously differentiable as required by hypotheses of Theorem 2.1, one can
write ν(ϑ, B) = Y (ϑ, B)ν(ϑ, B) + (1 − Y (ϑ, B))ν(ϑ, B), η(ϑ, B) = Y (ϑ, B)η(ϑ, B) + (1 − Y (ϑ, B))η(ϑ, B), (164)
where Y ∈ D(R2) is a suitable function.
Now we have ν(ϑδ, Bδ) huδi + η(ϑδ, Bδ)divx(uδ)I = Y (ϑδ, Bδ)ν(ϑδ, Bδ) huδi + Y (ϑδ, Bδ)η(ϑδ, Bδ)divx(uδ)I+ n (1 − Y (ϑδ, Bδ))ν(ϑδ, Bδ) huδi + (1 − Y (ϑδ, Bδ))η(ϑδ, Bδ)divx(uδ)I o ,
where the expression in the curl brackets can be made arbitrarily small in the norm of Lp((0, T ) × Ω),
with a certain p > 1, by a suitable choice of Y (see estimates (112), (119), (125) and formula (133). Thus relation (163) holds for any ν and η satisfying hypotheses of Theorem 2.1
3. The oscillations defect measure. In order to describe oscillations in the sequence {̺δ}δ>0, we use
the renormalized continuity equation (59) together with its counterpart resulting from letting δ → 0 in (156).
Although we have already shown that the limit quantities ̺ and u satisfy the momentum equation (57), the validity of its renormalization (59) is not clear as the sequence {̺δ}δ>0 is not known to be
uniformly square integrable and the Di Perna-Lions machinery [28] does not work a priori. In order to solve this problem, a concept of oscillations defect measure was introduced in [38]. To be more specific we set
oscq[̺δ→ ̺](Q) = sup k≥1 lim sup δ→0 Z Q|T k(̺δ) − Tk(̺)|q dx dt , (165)
where Tk are the cut-off functions
Tk(z) = sgn(z) min{|z|, k}.
• ̺δ, uδ satisfy (156),
• {uδ}δ>0 is bounded in L2(0, T ; W1,2(Ω)),
•
oscq[̺δ → ̺]((0, T ) × Ω) < ∞ for a certain q > 2. (166)
Then, in order to establish (59) it is clearly enough to show (166).
4. Weighted estimates of the oscillations defect measure. In order to prove (166) we use weighted estimates, where the corresponding weight function depends on the absolute temperature.
Taking b = Tk in (163) we get p(̺, ϑ)Tk(̺) − p(̺, ϑ) Tk(̺) (167) =4 3ν(ϑ, B) + η(ϑ, B) divx(u) Tk(̺) − 4 3ν(ϑ, B) + η(ϑ, B) divx(u) Tk(̺).
As observed in (45), there is a positive constant pcsuch that pF(̺, ϑ) − pc̺
5 3 is a non-decreasing function of ̺ for any ϑ. Accordingly we have p(̺, ϑ)Tk(̺) − p(̺, ϑ) Tk(̺) ≥ pc ̺53Tk(̺) − ̺ 5 3 Tk(̺) . (168)
Now let us choose a weight function w ∈ C1[0, ∞)
w(ϑ) > 0 for ϑ ≥ 0, w(ϑ) = ϑ−1+α2 for ϑ ≥ 1, (169)
where α is the exponent appearing in hypothesis (54). Multiplying (167) by w(ϑ) and using (168) we obtain
w(ϑ)̺53Tk(̺) − ̺ 5 3 Tk(̺) (170) ≤ w(ϑ) pc 4 3ν(ϑ, B) + η(ϑ, B) divx(u) Tk(̺) − w(ϑ) pc 4 3ν(ϑ, B) + η(ϑ, B) divx(u) Tk(̺),
where the right-hand side is a weak limit of w(ϑδ) pc 4 3ν(ϑδ, Bδ)+η(ϑδ, Bδ) divx(uδ) Tk(̺δ)−Tk(̺) +w(ϑ) pc 4 3ν(ϑ, B)+η(ϑ, B) divx(u) Tk(̺)−Tk(̺) . Consequently, using the uniform weighted estimates (119) together with hypothesis (19) and the growth restriction on w specified in (169), we conclude that
Z T 0 Z Ω hw(ϑ) pc 4 3ν(ϑ, B)+η(ϑ, B) divx(u) Tk(̺)− w(ϑ) pc 4 3ν(ϑ, B)+η(ϑ, B) divx(u) Tk(̺) i dx dt (171) ≤ c lim inf δ→0 kTk(̺δ) − Tk(̺)kL 2((0,T )×Ω), with c independent of k.
On the other hand, it was shown in Chapter 6 in [39] that the left-hand side of (170) could be bounded below as Z T 0 Z Ω w(ϑ)̺53Tk(̺) − ̺53 Tk(̺) dx dt ≥ lim sup δ→0 Z T 0 Z Ωw(ϑ)|T k(̺δ) − Tk(̺)| 8 3 dx dt. (172)
Then (170) together with (171), (172), give the weighted estimate
lim sup δ→0 Z T 0 Z Ωw(ϑ)|T k(̺δ) − Tk(̺)| 8 3 dx dt ≤ c lim inf δ→0 kTk(̺δ) − Tk(̺)kL 2((0,T )×Ω). (173)
Now taking q > 2 and 8 3qω =
1+α
2 , we can use H¨older’s inequality to obtain
Z T 0 Z Ω|T k(̺δ) − Tk(̺)|q dx dt = Z T 0 Z Ω|T k(̺δ) − Tk(̺)|q(1 + ϑ)−ω(1 + ϑ)ωdx dt (174) ≤ c Z T 0 Z Ω|T k(̺δ) − Tk(̺)| 8 3(1 + ϑ)− 1+α 2 dx dt + Z T 0 Z Ω (1 + ϑ)3q(1+α)2(8−3q) dx dt .
On the other hand, in accordance with (112), (125) ϑ ∈ L∞(0, T ; L4(Ω)) ∩ L3(0, T ; L9(Ω)), and a simple
interpolation argument yields ϑ ∈ Lr((0, T ) × Ω), with r = 46
9. Consequently, one can find q > 2 such
that
3q(1 + α) 2(8 − 3q) = r =
46 9 , provided α complies with hypothesis (70).
Then finally relations (173), (174) give rise to the desired estimate
lim sup δ→0 Z T 0 Z Ωw(ϑ)|T k(̺δ) − Tk(̺)|q dx dt ≤ c 1 + lim inf δ→0 kTk(̺δ) − Tk(̺)kL 2((0,T )×Ω) , yielding (166).
5. Propagation of oscillations and strong convergence. According to (166) we know that the renor-malized continuity equation (59) is satisfied by the limit functions ̺ and u. In particular
∂tLk(̺) + divx(Lk(̺)u) + Tk(̺) divx(u) = 0 in D′((0, T ) × R3), (175)
provided ̺, u have been extended to be zero outside Ω, where Lk(̺) = ̺
Z ̺
1
Tk(z)
z2 dz .
On the other hand, one can let δ → 0 in (156) and one obtains
∂tLk(̺) + divx(Lk(̺)u) + Tk(̺) divx(u) = 0 in D′((0, T ) × R3). (176)
Now, following step by step Chapter 3 in [42] we take the difference of (175) and (176) and use (167) and (168) to deduce Z Ω Lk(̺)(τ ) − Lk(̺)(τ ) dx ≤ Z τ 0 Z Ω (divx(u)) Tk(̺) − Tk(̺) dx dt for any τ ∈ [0, T ],
where the right-hand sides vanishes for k → ∞ due to (166). Consequently
̺ log(̺) = ̺ log(̺) in (0, T ) × Ω,
a relation equivalent to strong convergence of {̺δ}δ>0, which means in fact that
̺δ → ̺ in L1((0, T ) × Ω). (177)
2.2.7 End of the proof
Note that we have already shown that ̺, u satisfy the continuity equation (57) as well as its renormalized version (59). Moreover, it is easy to see that the “averaged” momentum equation (147) coincides in fact with (60) in view of the strong convergence results established in (146), (160) and (177). By the same token, one can pass to the limit in the energy equality (108) in order to obtain (67). Note that the regularizing δ−dependent terms on the left-hand side disappear by virtue of the estimates (111), (112), and (136).
Similarly, one can handle Maxwell’s equation (107). Here, the only thing to observe is that the terms 1
σ(̺δ, ϑδ, Hδ)
curlxHδ are bounded in Lp((0, T ) × Ω), for a certain p > 1,
uniformly with respect to δ (such a bound can be obtained exactly as in (133).
To conclude, we have to deal with the entropy inequality (106). First of all, the extra terms on the left-hand side tend to zero because of (111), (112), (130) and (151). Furthermore, it is standard to pass to the limit in the entropy production rate keeping the correct sense of the inequality as all terms are convex with respect to the spatial gradients of u, ϑ and B.
Finally, the limit in the logarithmic terms can be carried over thanks to estimate (130) and the following result (see Lemma 5.4 in [29] holds
Lemma 2.2 Let Ω ⊂ RN be a bounded Lipschitz domain. Assume that
ϑδ → ϑ in L2((0, T ) × Ω) and log(ϑδ) → log(ϑ) weakly in L2((0, T ) × Ω).
Then ϑ is positive a.e. on (0, T ) × Ω and log(ϑ) = log(ϑ).
3
Other models
3.1
Polytropic models
The magnetic Navier-Stokes-Poisson case where magnetic permeability µ is a pure positive constant was studied by Ducomet and Feireisl in [30].
In the absence of magnetic field a model of pressure law more adapted to cold plasmas of the type p(̺, ϑ) = pe(̺) + ϑpϑ(̺), (178)
with associated internal energy e(̺, ϑ) such that
eρ=
1
ρ2(p − θpθ), eϑ= CV(ϑ), (179)
introduced by Feireisl [39] in the case of constant viscosities and generalized to temperature dependent vis-cosities and conductivity in [40], has been extended to the MHD case (with Dirichlet boundary condition
B|∂Ω= 0 and constant resistivity) by Hu and Wang in [51]:
Theorem 3.1 Let Ω ⊂ R3 be a bounded domain with boundary of class C2+δ, δ > 0.
Suppose that the pressure p = p(̺, ϑ) is given by (178), that pe, pϑ are C1functions of their arguments
on [0, ∞) such that
pe(0) = pϑ(0) = 0,
p′e(ρ) ≥ a1ργ−1, pϑ(ρ) ≥ 0 for all ρ > 0,
pe(ρ) ≤ a2ργ, pϑ(ρ) ≤ a3(1 + ργ/3) for all ρ ≥ 0.
Furthermore, we suppose that the magnetic permeability µ and the electrical conductivity σ are positive constants and that the transport coefficients
κ = κ(ϑ), ν = ν(ϑ), 1
are continuously differentiable functions of their argument satisfying c1(1 + ϑα) ≤ κ(ϑ) ≤ c2(1 + ϑα), (180) 0 ≤ λ(ϑ) ≤ λ, (181) 0 < ν≤ ν(ϑ) ≤ ν, (182) and 0 < cV ≤ cV(ϑ) ≤ cV, (183) for c1, c2, λ, ν, ν, cV, cV > 0 and α > 2.
Finally assume the homogeneous boundary conditions
H|∂Ω= 0, u|∂Ω= 0, q · n|∂Ω= 0,
and let initial data ̺0, (̺u)0, ϑ0, B0 be given so that
̺0∈ L 5 3(Ω), (̺u)0∈ L1(Ω; R3), ϑ0∈ L∞(R3), B0∈ L2(Ω; R3), (184) ̺0≥ 0, ϑ0> 0, (185) (̺s)0= ̺0s(̺0, ϑ0), 1 ̺0|(̺u)0| 2, (̺e) 0= ̺0e(̺0, ϑ0) ∈ L1(Ω), (186) and divx B0= 0 in D′(Ω), B0· n|∂Ω= 0. (187)
Then problem (57)-(65) possesses at least one variational solution ̺, u, ϑ, B on an arbitrary time interval (0, T ).
This model, with boundary conditions
B · n|∂Ω= 0, curlxB × n|∂Ω= 0,
has been extended to the variable conductivity case
0 < σ ≤ σ(̺, ϑ) ≤ σ,
for constants σ, σ > 0 and for α ≥ 2, but under additional constraints on viscosities, by Fan and Yu [36].
3.2
Barotropic models
When the temperature fluctuations can be neglected, one obtains the barotropic-MHD system for the density ̺, the velocity u and the magnetic field H in a bounded region Ω
∂t̺ + divx(̺u) = 0, (188)
∂t(̺u) + divx(̺u ⊗ u) = divxT+ j × B, (189)
∂tH + curlx(H × u) + curlx(λcurlxH) = 0, divxH = 0, (190)
u|∂Ω= 0, H|∂Ω= 0. (191)
where the pressure is p(̺) = A̺γ and the transport coefficients ν, η, λ are positive constants.
Theorem 3.2 Let Ω ⊂ R3 be a bounded domain with boundary of class C2+δ, δ > 0 and γ > 3/2.
Assume that the initial data ̺0, (̺u)0 and B0 be given so that
̺0∈ Lγ(Ω), (̺u)0∈ L1(Ω; R3), H0∈ L2(Ω; R3), (192)
̺0≥ 0, (193)
and
divxH0= 0 in D′(Ω), H0|∂Ω= 0. (194)
Assume finally that f, g ∈ L∞(0, T × Ω).
Then problem (188)-(191) possesses at least one variational solution ̺, u, H on an arbitrary time interval (0, T ).
Let us mention that the same result holds (R. Sart [86]) with different boundary conditions on H. Let us quote for completeness some other models recently introduced.
Y. Amirat and K. Hamdache [1] studied a barotropic ferrofluid model with a magnetization dependent pressure p = p(̺, M) including in the fluid description an extra equation for the angular momentum.
Finally X. Hu and D. Wang [53] considered density dependent viscosities, extending to MHD the result of D. Bresch and B. Desjardins [10] relying on analysis of the so-called BD-entropy. In this last case, presence of the radiative Stefan-Boltzmann contribution ϑ4 is no-more requested but additional
constraints on the pressure and transport coefficients (including magnetic permeability) are needed.
4
Singular limits
As usual in hydrodynamical problems, when putting the system in non dimensional form, various non dimensional numbers [101] (Mach, Reynolds, Peclet etc...) appear, which may be small (or large) in certain regimes of motion. Studying the corresponding limits may simplify drastically the description. Typically low Mach number limits of compressible flows are expected to be close to their incompressible limit. In MHD flows, asymptotic regimes appear naturally when the so called Mach number, Alfven number, Froude number or P´eclet number are small.
The simplest case has been studied by P. Kukuˇcka [65], for which one expects convergence toward an incompressible limit.
More precisely for a suitable scaling (see [65]) one considers the following non dimensional form of the MHD system (9)-(14)-(17)-(24) where magnetic permeability µ is supposed to be a positive constant (and we note λ(̺, ϑ, H) := µσ(̺,ϑ,H)1 the magnetic diffusivity), to simplify the exposition.
∂tB + curlx(B × u) + curlx(λcurlxB) = 0, (195)
∂t̺ + divx(̺u) = 0, (196)
∂t(̺u) + divx(̺u ⊗ u)
1 M a2 = divxS+ 1 Al2j × B + 1 F r2̺∇xΦ, (197) d dt M a2 2 ̺|u| 2+ ̺e +M a2 Al2 1 2µ|B| 2dx = 0, (198) ∂t(̺s) + divx(̺su) + 1 P edivx q ϑ = σ, (199) with σ ≥ ϑ1M a2S: ∇xu + M a2 Al2 |curlxH| 2 −P e1 q · ∇ϑxϑ, (200) where Φ is an external potential (gravitation for example).
In order to simplify the exposition, following [65], we suppose that Mach number M a and Alfven number Al are both small M a = Al = ε with 0 < ε ≪ 1 while P e = 1. We also discard the external potential force (Φ = 0).
We assume boundary conditions
B · n|∂Ω= 0, E × n|∂Ω= 0, u · n|∂Ω= 0, Sn × n|∂Ω= 0, ∇xϑ · n|∂Ω= 0, (201)
and initial conditions
(B, ̺, u, ϑ)(0, x) = (B0, ̺0, u0, ϑ0)(x) for any x ∈ Ω, (202)
“ill-prepared” in the sense that
B0(x) = B(1)0,ε(x), ̺0(x) = ̺ + ε(1)0,ε(x), u0(x) = u(1)0,ε(x), ϑ0(x) = ϑ + ϑ(1)0,ε(x), (203)
where ̺ and ̺ are two positive constants.
Formally expanding (195)-(199) with respect to ε we actually find at lowest order the incompressible MHD system with temperature
∂tB + curlx(B × U) + curlx λ(̺, ϑ, B)curlxB = 0, (204)
divx(U) = 0, (205)
̺ (∂tU + U · ∇xU) + ∇xΠ = divx S(U, ϑ) + curlxB × B, (206)
̺cP(̺, ϑ) (∂tΘ + U · ∇xΘ) = divx κF(̺, ϑ, B) + κrϑ 2 ∇xΘ , (207) where cP(̺, ϑ) = eϑ+̺ϑ2 p2ϑ
p̺, with boundary conditions
B · n|∂Ω= 0, U · n|∂Ω= 0, ∇xΘ · n|∂Ω= 0, (208)
and initial conditions
B(0, x) = B0(x), U(0, x) = u0(x), Θ(0, x) = ϑ0(x) for any x ∈ Ω. (209)
then one has the convergence result [65]
Theorem 4.1 Let Ω ⊂ R3 be a bounded domain with boundary of class C2+δ, δ > 0 and assume that
properties of the state functions and transport coefficients given in Theorem 2.1 are valid.
Let {̺ε, uε, ϑε, Bε}ε>0be a family of weak solutions to the scaled MHD system (195)-(199) on (0, T )×Ω
supplemented with boundary conditions (201) and initial conditions (203) such that ̺0,ε→ ̺(1)0 weakly* in L∞(Ω), u0,ε→ u(1)0 weakly* in L∞(Ω; R3), ϑ0,ε→ ϑ(1)0 weakly* in L∞(Ω), B0,ε→ B(1)0 weakly* in L∞(Ω; R3), as ε → 0. Then ess supt∈(0,T )k̺ε(t) − ̺kL5 3(Ω)≤ Cε, uε→ U weakly in L2(0, T ; W1,2(Ω; R3), Bε ε → B weakly in L 2(0, T ; W1,2(Ω; R3),
ϑε− ϑ
ε → Θ weakly in L
2(0, T ; W1,2(Ω; R3),
where (U, Θ, B) is a weak solution of the incompressible MHD system with temperature (204)-(207) with boundary conditions (209), initial temperature
Θ0= ϑ cP(̺, ϑ) s̺(̺, ϑ)̺(1)0 + sϑ(̺, ϑ)ϑ(1)0 ,
and initial velocity given by
̺U(0) = lim
ε→0H[̺εuε](0),
where H is Helmholtz’s projector on the space of solenoidal fields.
The proof follows the stategy of [42]:
1. One first proves uniform estimates for the sequence {̺ε, uε, ϑε, Bε}ε>0 and the entropy rate σε.
2. One pass to the limit ε → 0 in the system, the most delicate term being the convective term ̺εuε⊗ uε, for which one uses the Helmholtz decomposition together with the associated acoustic
equation.
More complex situations have been investigated: A. Novotn`y, M. R˚uˇziˇcka ang G. Th¨ater [81] have considered the case M a = ǫ, Al = ǫ, F r = ǫ, P e = ǫ2, Y.S. Kwon and K. Trivisa [68] have studied the
case M a = ǫ, Al = √ǫ, F r = √ǫ, P e = 1, E. Feireisl, A. Novotn`y and Y. Sun [43] have considered M a = ǫ, Al = F r = P e = 1 Re = 1/ǫn, for a n > 0 (large Reynolds number regime) and S. Jiang, Q. Ju,
F. Li and Z. Xin [56] have studied more complex ǫk regimes. In all of these situations the target system
is the incompressible MHD system.
5
Strong solutions
Previous Sections dealt with weak solutions, belonging to Sobolev spaces, for system (9)-(14)-(17)-(21), together with (7) and (8) for the definition of the electric field and the current density. It is not known in general if these solutions are unique.
Another approach is to look directly for strong solutions, which are assumed to be regular. In order to do so, the general method is to prove local existence of smooth solutions. This is done by applying general local existence results on hyperbolic and hyperbolic-parabolic systems (Subsection 5.1 below). Then, it is possible, assuming that the initial data are sufficiently close to an equilibrium state, to prove that the parabolic nature of the equation dominates. This implies a priori bounds which allow to prove that the solution remains in the neighbourhood of the equilibrium as long as it exists. This in turn allows to prove global existence. This method, due to Kawashima and his co-authors, is described in Subsection 5.2. Subsection 5.3 is devoted to existence and uniqueness results in critical (Besov) spaces.
5.1
Local existence results
In [95], it is proved that a symmetric hyperbolic-parabolic system has a unique local solution. Therefore, its application to the present system only amounts to proving that the system is symmetrizable. The use of entropy variables is necessary to give a symmetric system for the fluid part (equations (14)-(17)-(21)). See for instance [21] for the details. The part with the magnetic field is dealt with using the method of [87], so the results of [95] apply. The result may be summarized as follows:
Theorem 5.1 Let Ω = R3, and assume that the thermodynamic functions
p = p(̺, ϑ), e = e(̺, ϑ), s = s(̺, ϑ)
are interrelated through (25), where p, e are continuously differentiable functions of positive arguments ̺, ϑ satisfying (34)-(35)-(36)-(37).
Furthermore, we suppose that each of the transport coefficients
ν = ν(ϑ, |H|), η = η(ϑ, |H|), κF = κF(̺, ϑ, |H|), σ = σ(̺, ϑ, |H|) and µ = µ(|H|),
are positive smooth functions of their arguments. Consider an equilibrium state ̺, u ≡ 0, ϑ, B ≡ 0 of the system (9)-(14)-(17)-(21), and let the initial data data ̺0, u0, ϑ0, B0 be given so that
̺0− ̺, u0, ϑ0− ϑ, B0 ∈ H3(R3). (210)
Then there exists T > 0 such that the system (9)-(14)-(17)-(21), with the initial condition ̺0, (̺u)0, ϑ0,
B0, has a unique strong solution on [0, T ]. This solution satisfies ̺, ϑ > 0 in R3× [0, T ], and T depends
only on
̺0− ̺, u0, ϑ0− ϑ, B0
H3, inf
R3ϑ0 and infR3 ̺0.
Some remarks are in order.
First, the above result is stated in [61], for electro-magneto-fluids (which, in addition to the above systems, includes equations for the charge ̺eand the electric field E) with two-dimensional symmetries.
However, the aim of [61] is to prove global existence, and this is why their study is restricted to two-dimensional flows. The local existence result is still valid for the 3D case. As a matter of fact, it is an application of the results of [95], which are valid in any dimension.
Second, the same result can be stated with Hsinstead of H3, for any s ≥ 3 (see [95, Section 4]).
Third, Theorem 5.1 is proved in [84, Theorem 2.1] in the special case of constant coefficients ν, η, κF, σ
and µ. The proof is not given, and is claimed to be an easy adaptation of that of [77]. Looking at the proof of [77], it is restricted to the case B = 0, that is, the compressible Navier-Stokes system with thermal conductivity. Their proof readily applies to the present case, and allows in fact for variable coefficients satisfying the above assumptions. A complete proof is also given in [37], in the case of a bounded domain (the method is easily adapted to Ω = R3), with possible occurrence of vacuum, that is, ̺ ≥ 0 instead of
̺ > 0.
Note also that, in Theorem 5.1, we have assumed (34), that is, the fluid is a perfect gas. Now, the result is still valid for a general equation of state satisfying (35)-(36)-(37) only (see [61]).
Finally, the assumptions ν, η, κF, σ, µ > 0 may be weakened in the following way: it is in fact possible
to treat cases in which some of these coefficients are identically 0. For instance, using the same method as in [61], it is possible to deal with the case ν, η, σ, µ > 0, κF ≡ 0, the case ν, κF, σ, µ > 0, η ≡ 0, and the
case ν, σ, µ > 0, κF, η ≡ 0.
5.2
Global existence results for small data
Theorem 5.1 being proved, one can then turn to the proof of global existence of a smooth solution for the system under consideration. For this purpose, the key ingredient is an a priori estimate which we now detail. For this purpose, we define, for any 0 ≤ t1≤ t2, the quantity
Ns(t1, t2) = sup t1≤t≤t2 ̺ − ̺, u, ϑ − ϑ, B (t) Hs(R3) + Z t2 t1 k(∇̺, ∇B)k2Hs−1(R3)(t) + k(∇u, ∇ϑ)k 2 Hs(R3)(t) dt 1/2 . (211) Then, differentiating the system if necessary, and multiplying it by some well chosen test functions, it is possible to prove the following kind of estimate: there exists δ > 0 such that, if
